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I'm learning about tautologies right now. I see that a tautology is when two propositional statements have the same truth values. But I'm struggle with the truth table my professor provided about the tautology between "p implies q" and "not p or q"

The first two columns of the truth table are obviously the propositional variables, p and q. The next column represents "not p" which I don't understand why we'd have that considering the first column contains instances of "not p". The fourth column is where I become confused, because from it I get the conclusions, "not p or q" is true when both p and q are true (which I agree with), it is false when p is true and q is false (which I don't understand. I thought as long as at least one propositional variable is true within a disjunction the statement is true as long as its not the exclusive or?, true when p is false and q is true (this makes sense, that's the proposition itself), and true when p is false and q is false (again, I don't understand this. Here we have a disjunction where both variables are false, and yet somehow the proposition "not p or q" is true?)

Can someone explain what I'm missing?

Lex_i
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    This question has been asked and answered many times in various flavors across the SE network. See [here](https://philosophy.stackexchange.com/questions/34082/why-are-conditionals-with-false-antecedents-considered-true), [here](https://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr) [here](https://math.stackexchange.com/questions/48161/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-false), ... – Natalie Clarius Sep 20 '20 at 15:42
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    ..., [here](https://philosophy.stackexchange.com/questions/36117/what-does-the-truth-value-of-a-material-implication-represent), [here](https://philosophy.stackexchange.com/questions/36746/why-does-the-material-conditional-have-the-truth-table-it-does) and [here](https://math.stackexchange.com/questions/232309/how-to-interpret-material-conditional-and-explain-it-to-freshmen). – Natalie Clarius Sep 20 '20 at 15:42

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